The main problem is that there are few if any flights from Heathrow direct to Dalaman. Most of the flights to Turkey are scheduled to Istanbul and therefore very expensive.
Our 2 Springer Spaniels flew out with us on a MyTravel flight from Bristol to Dalaman in August.As we entered the baggage hall the dogs were having aterrific time going round the carousel in their travelling boxes. Their boarding and paperwork, including legalisation, was all taken care of by Overhill Kennels of Bristol.They also supplied their travelling boxes. If you use any of the kennels in the London catchment area you'll pay a high premium. Overhill have got a website that is well worth checking out. You must travel on the same flight as the dog, otherwise an import licence is required.
They're loving it here in Turkey. Although you'll find that not as many Turks are as keen on dogs as the English are. In fact quite a few Turks that we've met have been very afraid of them until they've got to know them. ( We actually managed to clear the water of Turks at Gemiler when we took the dogs for a swim ).
Contact me if you need info regarding airport transfer from Dalaman, because there are very few taxi companies that are prepared to carry animals. We may also be able to help regarding the dog sitting / kenneling in the Fethiye area.